Character Analysis
Captain Styles is the commanding officer of the USS Excelsior, a foil for Admiral Kirk…and a professional jerkface. We're going to focus on the foil aspect of his character here.
Kirk and Styles both love their respective ships but for different reasons. Kirk loves the Enterprise because of the people and memories he associates with the ship. For him, the Enterprise is home. He even refers to it as a "house with all the children gone" at the beginning of the film.
Styles loves the Excelsior because it's a great bit of tech. He's calls it an "incredible machine" and says he'll use it "to [break] some of the Enterprise's speed records." For him, the Excelsior is an impersonal machine that will bring him status within Starfleet, not a home.
This extends beyond the ships to their crews. Whereas Kirk refers to his crew by their names and talks and laughs with them, Styles never refers to anyone by name and barks orders. Whereas Kirk knows his crew as individuals, Styles does not.
Oops: guess we're also focusing on the "professional jerkface" part of Styles' character.
In a film about friendship and loyalty, this is a bad example to set, and Styles gets his comeuppance, courtesy of one Mr. Scott. While working on the Excelsior, Scotty takes some rather important looking parts from the transwarp computer. When Styles gives chase to the Enterprise, his starship breaks down in front of Spacedock, the galactic equivalent of the driveway. The Excelsior's maiden voyage is a complete bust.
Of course, we know the question you really came here to have answered: What exactly is that baton thing Styles carries around? It's called a swagger stick. Not only were swagger sticks an actual thing, but the idea behind them provide the perfect accessory for a character like Styles.
You know: a professional jerkfa—we mean foil.